Elsevier

Mechanics of Materials

Volume 40, Issues 4–5, April–May 2008, Pages 407-417
Mechanics of Materials

Ordering effect of kinetic energy on dynamic deformation of brittle solids

https://doi.org/10.1016/j.mechmat.2007.10.003Get rights and content

Abstract

The present study focuses on the plane strain problem of medium-to-high strain-rate loading of an idealized brittle material with random microstructure. The material is represented by an ensemble of “continuum particles” forming a two-dimensional geometrically and structurally disordered lattice. Performing repeated lattice simulations for different physical realizations of the microstructural statistics offers possibility to investigate universal trends in which the disorder and loading rate influence mechanical behavior of the material. The dynamic simulations of the homogeneous uniaxial tension test are performed under practically identical inplane conditions although they span nine decades of strain rate. The results indicate that the increase of the dynamic strength with the loading-power increase is also accompanied with a significant reduction of the strength dispersion. At the same time, increase in the loading rate results in transition from random to deterministic damage evolution patterns. This ordering effect of kinetic energy is attributed to the diminishing flaw sensitivity of brittle materials with the loading-rate increase. The uniformity of damage evolution patterns indicates an absence of the cooperative phenomena in the upper strain-rate range, in opposition to the coalescence of microcracks into microcrack clouds, which may represent the dominant toughening mechanism in brittle materials not susceptible to dislocation activities.

Introduction

The practical and scientific significance of deformation processes evolving at high strain rates is exceeded only by their complexity; the considered thermodynamic processes are non-stationary, non-local, and far from equilibrium. Despite the truly enormous progress in experimental analysis, there are inherent difficulties, if not limitations, when it comes to extreme loading rates (⩾107 s−1). It is especially difficult to characterize the damage evolution and failure mechanisms due to the difficulty in recovering tested samples. The numerical experiments presented in this study cover the strain-rate range from medium to high, which are commonly explored by the Hopkinson bar and plate-impact experiments. The former is performed (up to 103–104 s−1) under uniaxial stress conditions while the latter (up to 107 s−1) under uniaxial strain conditions. The objective of this investigation is to elucidate the effect of material micro-texture on fracture of dynamically loaded brittle materials. The numerical simulations of the uniaxial tension test, the most common of all mechanical tests for structural materials, appear to be a useful tool for extrapolation of the experimental results.

The idealized brittle material is approximated by a two-dimensional triangular lattice: a Delaunay simplical graph dual to the irregular honeycomb system of Voronoi polyhedra representing, for example, grains of a ceramic material (Krajcinovic, 1996).1 Grain boundaries, the most common examples of weak interfaces in brittle materials, are known to have a profound effect on their structure-sensitive properties such as the dynamic strength. The primary mechanism of damage evolution in polycrystalline ceramic materials is widely reported to be the intergranular microcracking. According to Lawn (1993), grain boundaries “are especially weak in ceramics because of the stringent directionality and charge requirements of covalent–ionic bonds”. Furthermore, as the regions of lower atomic density, grain boundaries act as sources of, and sinks for, structural defects. Finally, sintering and hot pressing at high temperatures followed by cooling causes residual stresses due to the anisotropy of grains (Davidge and Green, 1968, Curtin and Scher, 1990). Therefore, in the present lattice model, the microstructural texture is represented by a network of grain boundaries; the cracking is assumed to be intergranular; and the local stress and strain fluctuations on a scale shorter than the grain facet size are neglected (Krajcinovic, 1996). In more general terms, we may think of lattice models as “a discretization of higher than one-dimensional materials by a network of one-dimensional elements characterized by element constitutive equations, and a breaking condition”, (Jagota and Bennison, 1994). The model incorporates both variability and uncertainty in a straightforward manner. Variability, also termed randomness or aleatory variability, is the natural randomness in the process. Uncertainty, also termed epistemic uncertainty, is the uncertainty in the model; it is due to limited knowledge or limited availability of data or both. The mesoscale material texture is an example of the aleatory variability. The disorder may be topological (unequal coordination number), geometrical (unequal length of bonds), or structural (unequal strength and stiffness of bonds). The disorder is further enhanced by damage evolution, which is governed (to an extent depending on the deformation rate) by the local fluctuations of the energy barriers quenched within the material, and the local fluctuations of stress.

Finally, although the lattice models are used often in modeling the fracture behavior of inhomogeneous or multi-phase systems, it is important to recognize their deficiencies and limits of their applicability. The two-dimensional systems exhibit inherent topological limitations; and the extreme geometrical disorder, which is intentionally used in this study, causes additional side effects. Nonetheless, we believe that the most notable numerical artifacts of lattice models (Jagota and Bennison, 1994, Monette and Anderson, 1994) do not have a first-order effect on universal trends observed in this investigation, keeping in mind that we are dealing with relatively large geometrical and structural disorder and the nucleation-dominated damage evolution modes.

Section snippets

Details of the simulation

The approximation of a material by a particle lattice is inspired by the mesoscale morphology of a certain class of brittle materials, and successes of molecular dynamics method as a tool of solid mechanics (for overview of molecular dynamics, see, for example, Hoover, 1986, Allen and Tildesley, 1994). Within this framework, continuum can be defined as a collection of discrete elements, known as “continuum particles” (Wiener, 1983). Assuming that positions and momenta of particles are known in

Dynamic tensile strength

The stress–strain curves obtained for 30 realizations at strain rates 1 and 1 × 107 s−1 are presented in Fig. 4. Three qualitative observations can be made:

  • the response is nearly linear up to failure for moderate and the highest strain rates, but not for the transient strain rates characterized by the rapid strength increase;

  • the loading-rate increase results in increase of the dynamic tensile strength (σm);

  • the loading-rate increase results in decrease of the strength dispersion.

The maximum dynamic

Summary

The objective of this work is to investigate universal trends in which the disorder and strain rate influence dynamic behavior of the idealized brittle material, by performing repeated lattice simulations for different physical realizations of the microstructural statistics. The dynamic simulations of the uniaxial tension test, ε˙[0.1,1×108s-1], are performed under practically identical stress conditions. The results reveal the ordering effect of the kinetic energy on the dynamic response of

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